Signed degree sequences in signed multipartite graphs

A signed k-partite graph (signed multipartite graph) is a k-partite graph in which each edge is assigned a positive or a negative sign. If G(V1, V2, · · · , Vk) is a signed k-partite graph with Vi = {vi1, vi2, · · · , vini}, 1 ≤ i ≤ k, the signed degree of vij is sdeg(vij) = dij = d + ij − d − ij , where 1 ≤ i ≤ k, 1 ≤ j ≤ ni and d + ij(d − ij) is the number of positive (negative) edges incident with vij . The sequences αi = [di1, di2, · · · , dini ], 1 ≤ i ≤ k, are called the signed degree sequences of G(V1, V2, · · · , Vk). The set of distinct signed degrees of the vertices in a signed k-partite graph G(V1, V2, · · · , Vk) is called its signed degree set. In this paper, we characterize signed degree sequences of signed k-partite graphs. Also, we give the existence of signed k-partite graphs with given signed degree sets.